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Perception & Psychophysics1990. 47 (3), 227-242
Studies in auditory timing:2. Rhythm patterns
CAROLINE B. MONAHAN and IRA J, HIRSHCentral Institute for the Deaf, St. Louis, Missouri
Listeners discriminated between 6-tone rhythmic patterns that differed only in the delay ofthe temporal position of one of the tones. On each trial, feedback was given and the subject'sperformance determined the amount of delay on the next trial. The 6 tones of the patterns markedoff 5 intervals. In the first experiment, patterns comprised 3 "short" and 2 "long" intervals: 12121,21121, and so forth, where the long (2) was twice the length of a short (1). In the second experiment, patterns were the complements of the patterns in the first experiment and comprised 2shorts and 3 longs: 21212, 12212, and so forth. Each pattern was tested 45 times (5 positionsofthe delayed tone x 3 tempos x 3 replications). Consistent with previous work on simple interval discrimination, absolute discrimination (At in milliseconds) was poorer the longer the intervals (i.e., the slower the tempo). Measures of relative discrimination (At/t, where t was the shortinterval, the long interval, or the average of 2 intervals surrounding the delayed tone) were betterthe slower the tempo. Beyond these global results, large interactions of pattern with positionof the delayed tone and tempo suggest that different models of performance are needed to explain behavior at the different tempos. A Weber's law model fit the slow-tempo data better thandid a model based on positions of "natural accent" (Povel & Essens, 1985).
In the preceding study in this series on timing discrimination, Hirsh, Monahan, Grant, and Singh (1990) foundthat temporal interval discrimination for tonal onsets ina sequence of tones was similar to older results for thediscrimination of single temporal intervals (Abel, 1972;Creelman, 1962; Getty, 1975; Small & Campbell, 1962).The relative difference limen (DL; at/t) is fairly constantfor intervals from 100 to 1,500 msec, becomes larger forshorter intervals, and may even get better for longer intervals up to 8 sec (Killeen & Weiss, 1987).
A listener's accuracy in discriminating a small delayin the temporal position of a tone in a sequence is thusclearly related to the average interval separating the tonesor (reciprocally) to the tempo. Furthermore, when oneof the tones in a sequence is of a different pitch, or whenone of the intervals is made longer than the others, discrimination of timing is somewhat more difficult in thevicinity of the changed tone or interval (Hirsh et al.,1990). In the case of the longer interval, the change inperformance might reflect the same relative discrimination near a longer standard. No such explanation appliesto a pitch change, however.
In rhythmic structures, certain elements seem to be accented. Accents may be generated by pitch changes or
This research was supported by Air Force Grant AFOSR-87-0382to the Central Institute for the Deaf, and by Washington University,where Ira Hirsh is Distinguished University Professor. Thanks are extended to Julius Goldstein, Stephen Handel, and an anonymous reviewerfor comments on an earlier version of the manuscript. We especiallythank Joanne Kocunik and Jay Babcock for assistance in theexperiments.Correspondence may be addressed to Caroline B. Monahan or Ira J.Hirsh, Central Institute for the Deaf, 818 South Euclid, St. Louis, MO63110.
by longer temporal intervals, among other possibilities(Monahan & Carterette, 1985; Monahan, Kendall, &Carterette, 1987). Perhaps pitch changes and interval accents playa similar role. Are accented tones, from eithersource, easier or harder to discriminate temporally?
Accenting and PitchHirsh et al. (1990) observed that large pitch changes
had a more disruptive effect on timing discrimination thandid small ones. Might this have something to do with thefact that pitch intervals of more than 5 semitones (a musical fourth, or about 1/3 octave) are exceedingly rare inthe world's music, constituting less than 10% of intervalsin the literature surveyed (Dowling, 1968)? Dowling andHarwood (1986) have called this finding a musical universal, and suggest that it reflects a basic property of the auditory system. Tones that are more than 4 or 5 semitonesapart will tend to fall in different critical bands and willthus be less likely to mask one another. Fitzgibbons,Pollatsek, and Thomas (1974) have also reported that thedetection of temporal gaps between tones that were about2 112 octaves apart was very difficult. People notice gapswithin frequency streams or groupings rather thanbetweenstreams or groupings. Hirsh et al. expanded this resultby showing that at a relatively fast musical tempo(5 tones/sec or 300 beats/min), discrimination of the delay of a tone following a skip of 9 semitones was moredifficult than discrimination of the delay of tones in amonotone series or after a pitch skip of only 2 semitones.
It also appears that tones that skip more than 4 or 5 semitones sound accented at musical tempos and sound especially accented if the tone marks a point of inflection inthe pitch contour (Thomassen, 1982). Grant (1987) has
227 Copyright 1990 Psychonomic Society, Inc.
228 MONAHAN AND HIRSH
shown that a rise of Faof 4 semitones is sufficient to causea change in the perceived accent of a syllable in a sentence. Thomassen (1982), Monahan and Carterette (1985),and Monahan et al. (1987) have pointed out that tones thatbegin different auditory streams (Bregman & Campbell,1971; Dowling, 1968; van Noorden, 1975) at fast tempos(at 8 to 10 notes/sec) also tend to sound accented and beginnew pitch groupings at somewhat slower musical tempos.
Thus, both relatively large pitch skips and pitch-eontourinflections seem to reflect mechanisms of accent.
Accenting and Temporal IntervalsIf pitch changes lead to greater temporal uncertainty
by virtue of their establishing accents in a sequence, thenwe should be able to show similar effects through anothersource of accent. Among the various hypotheses aboutsources of accent, one depends solely on a change in timing. Tones in sequences appear to be accented (1) whenthe tone begins a long interval, and (2) to a lesser extent,when the tone ends a longer interval and begins a seriesof 3 or more intervals (povel & Okkerman, 1981). Asan explanation for the first sort of accent, Povel andOkkerman rely on what might be termed a "release frombackward masking" argument: the perceptual measurement of the relative strength of a tone takes time, andprocessing is interrupted when a new tone begins; therefore, the tone beginning a long interval is less subject tointerference by the following tone(s), and sounds accented. As to the second sort of accent, the authors offera "release from forward masking" argument: tones caninterfere with each other if the time between the offsetof one and the onset of the next is 250 msec or less; therefore, tones that end a lengthened interval and begin a newseries of 3 or more intervals are less subject to interference by the preceding tone(s), and may sound accented.The authors also suggest that there may be a "structural"basis, especially for the latter form of accent, that involvesthe hierarchical relations between beats and unaccentedpulses (see section below) that has been exemplified previously only in repeated or cycled rhythmic patterns.
Importance of the Regularity of AccentingOne of the hypotheses that we consider below assumes
that the listener carries a temporal referent of fixed length(a beat) against which to judge a single temporal changemade in a pattern. Therefore, we feel it is appropriate torecount briefly some of the evidence that suggests thatthe presence of regular accenting (accenting that occursat a fixed rate or interval) promotes both better perception and (re)production of auditory patterns. Martin (1972)first proposed that both speech and music could bedescribed as having a hierarchical scheme with equallydistributed accents, and showed in his experimental work(Shields, McHugh, & Martin, 1974; Sturges & Martin,1974) that auditory patterns characterized by such ascheme were recognized both more accurately and morequickly than patterns that were not characterized by sucha scheme.
Like Martin, other writers (Handel & Lawson, 1983;Handel & Oshinsky, 1981; Longuet-Higgins & Lee, 1982;Monahan, 1984; Monahan & Carterette, 1985; Monahanet al., 1987; Povel, 1984; Povel & Essens, 1985; Steedman, 1977; Yeston, 1976) have assumed that time inWestern music is psychologically arranged in a hierarchy as a rate or rates within a rate. That is, musical timemay be characterized as having a slower, periodic accentrate, a metric, "clock," or beat, that is internalized bythe listener. This clock is superimposed upon a temporalgrain-a string of equal intervals (some of which aremarked by note onsets and some of which are not).
Monahan and Carterette (1985, pp. 8-9) reviewed someexperimental work favoring Martin's hypothesis. Otherrecent studies showing the importance of the distributionof accenting have been performed by Povel (1981), Poveland Essens (1985), Boltz and Jones (1986), Monahanet al. (1987), and Fourakis and Monahan (1988). Thefindings of Povel (1981) are most directly relevant to ourexperiments. Povel first asked his subjects to reproduce(by tapping) different series of two alternating intervaldurations. Like Fraisse (1963), Povel found that the onlyratio of t, to t2 (where t) is short and t2 is long) that subjectsaccurately (re)produced was 1:2; other produced ratiostended to "drift" toward 1:2 or 1:1 ratios. Ratios of 1:3were very accurately produced only in limited contextsfor example, where there were 4 repeating intervals inthe ratios 1:1:1:3 (1113; e.g., 250-250-250-750 msec).Similarly, 1:4 ratios were accurately produced in therecycling pattern context 1:1:1:1:4 (11114; e.g.,250-250-250-250-1,000 msec). (In what follows, we willrefer to time patterns in terms of the number of temporalgrains or equal intervals between note onsets; thus, 1113and 11114 are patterns of interonset intervals, or 101patterns.) Povel (1981) suggested that what listeners hear inthe pattern 1113 is a recurring temporal accent every750 msec: one temporal accent is for the tone beginningthe long (3) interval and one is for the tone ending thelong interval that begins the run of the 4 intervals asdescribed in the preceding section. These accents demarcate 2 beat intervals: the first is filled with beat subdivisions that have a rate three times that of the beat rate,and the second beat interval is empty. On the other hand,Povel found that repeating patterns, such as 442 (e.g.,1,000-1,000-500 msec), were reproduced only with greatdifficulty and variability, presumably because the tone onsets that might be used to set up the perception of a regular beat in the listener are not equally distributed in timein such a pattern.
Western music favors the division of beats into two orthree equal parts. Fraisse (1982) surveyed more than 50pieces of Western classical music and found that most ofthe temporal values in any piece fell into two categories:long time values and short time values, where the longwas either 2 or 3 times longer than the short, but the shortoccurred 2 or 3 times, respectively, more often thanthe long. In our series of studies, in order to makeour patterns musically' 'sensible," we have restricted the
ratios of temporal intervals in our standard patterns to 1:1and 1:2.
Alternative Hypotheses for Timing DiscriminationIf, in a sequence, the temporal position of tones marking
the onset and offset of longer intervals is uncertain, onecan also invoke Weber's law for explanation, as was doneby Hirsh et al. (1990). There may be, however, still otherfeatures of the sequence, such as groupings, runs, andso forth, which do not necessarily point to accents yet maybe related to temporal uncertainty. Our experiments aredesigned to afford an opportunity to test these variousnotions as they apply to timing discrimination.
PurposesAccent versus the size of intervals. One main ques
tion, then, is whether, at musical tempos, the discriminability of delays of tones in rhythmic patterns can be betterinterpreted as a function of the position and regularity of"natural temporal accent" (povel & Essens, 1985) or,alternatively and more parsimoniously, as a function ofthe size of neighboring intervals in a pattern, as wouldbe suggested by Weber's law.
Hirsh et al. (1990, Experiment 3) also raised this question, but were unable to suggest an answer, because, forthe simple rhythmic sequences they employed, a modelbased on Weber's ratio and one based on the position ofnatural temporal accent would make almost exactly thesame predictions with regard to temporal discriminability. The patterns we employ in the present experimentsare sufficiently complicated to enable a choice betweenthe predictions of an accent model or an interval-length(Weber) model.
Choice of procedure. A second question raised byHirsh et al.'s (1990) Experiment 3 was one of procedure.How can we best measure absolute and relative temporalDLs at particular positions in a pattern? In our series ofstudies, the delay of a single tone relative to its positionin a standard pattern concomitantly lengthens the interval preceding the tone and shortens the interval following the tone. Either interval may subserve the measureof absolute discriminability (in milliseconds). Note thatour "delay-only" procedure differs from the "single gapdiscrimination" of Pollack (1967) and Bharucha and Pryor(1986), where an increase in the size of a single intervalrelative to the size of the corresponding interval in thestandard sequence concomitantly changes the length ofthe sequence.
Measures of relative discrimination. A third relatedquestion raised by Hirsh et al.'s (1990) Experiment 3 concerns the measure of relative discrimination: What is theappropriate t in the ratio !:l.t/t when there are two different temporal intervals in a sequence? Should t be the shortor the long interval? Or, in order to control for differingnote densities across patterns in different studies, shouldit be the average length of note intervals in a pattern? Or,since in our series of studies a single delayed tone
TIMING IN RHYTHMIC PATTERNS 229
lengthens the preceding interval and concomitantlyshortens the following interval, should t be based on thepreceding interval, the following interval, or an averageof the two? The first set of choices, using short, long,or average pattern interval as t, implies that the listenercarries a fixed temporal referent against which to judgea delay at any point in the pattern. The second set ofchoices, using the preceding or following interval, or theaverage of the 2 intervals surrounding the delay, impliesa process model of judgment that is dependent on the sizeof the intervals relative to the point of delay within thesequences. We will view our results with examples of bothtypes of referent.
Unitization. A fourth concern of this paper is measuring temporal DLs in patterns at tempos where Weber'slaw is known to break down for single-interval discrimination. Tones seem to lose their individual quality at apresentation rate of lO/sec or faster (as in Pollack, 1967).If a pattern of tones is played at a high enough rate, thepattern itself is heard as a unit-a chunk of sound thatcomprises the whole cycle. Both Handel and Oshinsky(1981) and Royer and Robin (1986) reported that perceivedunitization of repeating patterns occurred when the smallestIOI was about 100 msec (or a rate of 10 tones/sec). Thus,we will measure temporal DLs both above and below thisrate within different patterns.
Nature of the task and the amount of training. Twoother concerns are more general in nature. The discriminability of temporal differences has typically been measuredby the tapping responses of the subject. It has been arguedthat production or reproduction of rhythm patterns is basedon perceptual and memory schema of temporal patterns(Povel, 1981). While this may be the case, productionalso involves the motor system. Thus, we have chosenhere to have our subjects perform discrimination tasks thatbypass the motor limitations that a tapping task may impose. Second, since we are interested in the limitationsof the auditory system, we are primarily concerned withwhat listeners can do after a great deal of training andnot what they may do on the spur of the moment. Furthermore, we try to present stimuli under optimum conditionsfor discriminating delay. In practice, this has meant thatbefore every adaptive run of trials, the subjects are informed of the serial position of the tone to be delayed andthey have indicated that they could hear a delay equivalent to that presented on the first trial of each run.
In two experiments, we explored the dependence of alistener's temporal precision on features of monotone sequences. In both experiments, we restricted the patternsto 6 tones separated by unequal intervals.
EXPERIMENT 1
PatternsIn Experiment 1, eight patterns had 2 long and 3 short
intervals between 6 successive tones; a ninth pattern,11211, had also been employed by Hirsh et al. (1990, Ex-
230 MONAHAN AND HIRSH
POSITION OF FIRST LONG INTERVALDISTANCEBETWEEN"LONGS"
ONE TVO Ttl'EE
s s < ~ s s < s < s ~ sZERO L!Jll± UJJJ± uuu"SHORTS"
s s < ~ s s ~ < s s sONE LUJJ± lJJJJi ~"SHORT"
s < s ~ s < s ~ < i < So
TVO lJ±JJ± lt1ltl l±lJJi"SHORTS"
Figure 1. The nine standard 6-tone patterns of Experiment 1. Patterns are shown on an equal-interval tirneline andare denoted by the relative sizes of their 5 interonset intervals. Eight patterns comprise 2 long and 3 short intervals. Theninth pattern, 11211, is included as a control. Lo~ (2s) are twice the length of shorts (Is). A final interval after thesixth tone was reported by subjects to be long. Levels of "natural accenting" are shown for each tone according to therules of Povel and his coUeagues (see text): strong temporal accent is denoted as :s, moderate accent is denoted as <,and the weakest level of accent is not marked. Patterns are arranged in the figure verticaUy by distance between the 2long intervals and horizontally by position of the first short interval (see text).
periment 3) and was included here as a control to aid usin comparing results across the two studies. "Long" (2)is twice the length of "short" (1).
Figure 1 shows the patterns used in Experiment 1; toneonsets are shown on an equal-interval timeline.
These patterns are not only more complex thanisochronic examples, but also contain a different temporalaccent structure while comprising (with the exception ofthe 11211 pattern) the same number and kinds of intervals. We have not recycled our patterns but rather havesampled 8 of 10 possible orders of the total set containing 3 shorts and 2 longs. There are two permutation setswithin the total set, each containing five members:
Permutation Set 1: 22111 12211 11221 1112221112.Permutation Set 2: 21211 12121 11212 21121 12112.
The first and second rows of Figure 1 show, respectively,the first three members of the two sets above. The thirdrow of Figure 1 shows the remaining two members ofSet 2 (21121 and 12112) as well as the 11211 pattern.In the first row, the 2 longs are adjacent; in the second,they are separated by a short; in the third, they are separated by 2 shorts. The columns in Figure 1 represent theserial position of the first long interval.
Figure 1 also shows the position of the natural temporalaccent within each sequence according to Povel andOkkerman (1981) and Povel and Essens (1985). Positionsof strong temporal accenting at the beginning of longer
intervals are marked with ":s;"; weaker accents for thetone at the end of a long interval that begins the run of3 or more intervals are marked by "<." We considerthe last tone in each sequence as accented because it appears to be followed by a long silent interval. Our subjects reported that these patterns seemed to include a finalsixth interval which they interpreted as long.
For patterns in the first and third rows, the most prevalent accenting occurs every 2 short (grain) intervals,whereas for patterns in the second row, accenting every3 grain intervals is just as likely as accenting every 2 grainintervals. The pattern 11211 has the most regular, equallydistributed temporal accenting every 2 grain intervals; afteran initial short interval, accenting for 12211 and 12112is also perfectly regular every 2 grain intervals. Accenting in 12121 is regular every 3 grain intervals. With theseensembles, we can study how temporal discrimination forone sound depends upon its position in the pattern-therelation between its position and the organization andlength of surrounding intervals. We may also determinewhether the chief dependency is on simpler notions ofWeberian discrimination or more sophisticated notions ofrhythmic simplicity and complexity such as those proposedby Povel and Essens (1985).
Overview of the DesignTemporal DLs (~t) within patterns of tones were mea
sured at three tempos where the lOis for short were 50,
TIMING IN RHYTHMIC PATTERNS 231
22
STRUCTURE OF TRIALEXAMPLE: PATTERN 22111 WITH DELAY OF THIRD TONE
STANDARD20 180 20 180 20 80 20 80 20 80 20 80 2000 msec
INTERPATTERNINTERVAL
......o 100 200 300 400 500 600 700 800
COMPARISON I20 180 20 20 80 20 80 20 80 20 80 2000 msec
INTERPATTERNINTERVAL
'- '-
INTERPATTERNINTERVAL
'-
o
2
100 200
180
2
300 400 500 600 700 800
COMPARISON 220 180 20 230 20*20 80 20 80 20 80 3000 msec
800700600500400300200100o
TER- RESPONSETTERN 2 I I INTERVAL
TERVAL"- "-/' --
INPAIN
Figure 2. Trial structure with the temporal pattern 22111 as the standard at the medium tempo.Two comparisons follow the standard, one of which contains a delayed tone relative to the tone'sposition in the standard (the second comparison in the example). Tone durations, silent intervals,interpattern intervals, and respo_ interval are indicated in milliseconds. In Comparison 2, the astemk(*) represents a 3O-msec silent interval.
100, or 200 msec and for long were 100,200, or 400 msec,respectively. Tones at Serial Positions 2-6 were delayedfor each of the nine patterns. Thus, the experiment maybe viewed as a complete factorial comprising 3 temposx 9 patterns x 5 positions of delay = 135 conditions.
MethodApparatus. A DEC PDP-l 173 computer controlled the experi
ment. Tones were lQOO-Hz sinusoids generated from a digital soundfile (20,000 samples/sec), passed to a D/A converter, filtered, andsent to earphones in an anechoic room. All tones were 20 msecin length (rise and fall times were 2 rnsec, sustained portion was16 rnsec). Steady-state equivalent sound level at the earphones wasmaintained at 80 dB SPL ± 0.5 dB in an artificial ear. Voltage fortones was calibrated daily. The computer collected subject responsesand stored them in data files along with stimulus and timingparameters for each trial.
Subjects. There were 4 subjects, 2 males and 2 females, whowere paid for their services. All subjects had normal auditory sensitivity. One subject (H.D.) had had 4 years of musical trainingand had also been a subject in a previous experiment in our laboratory. Another (J.B.) had had 5 years of musical training, was selfrun, and served as experimenter for most of the sessions of the othersubjects. The remaining subjects had had 5 and 9 years of musicaltraining, respectively.
Procedure. The subjects were tested singly and, in each replication of the experiment, performed 135 adaptive runs correspondingto the experimental conditions. At the beginning of each run, theexperimenter entered the pattern, tempo, and serial position of thetone to be delayed into the program that controlled the adaptive pro-
cedure. The experimenter told the subject which tone in the patternwould be delayed. The experimenter played the pattern first in itsstandard form and then with the initial delay, and ascertained thatthe subject could hear the difference. The initial delays at the slow,medium, and fast tempos were 50, 25, and 25 msec, respectively;these delays had been found to be easily detectable in pilot studies.
The experimenter then initiated the cued 2AFC procedure in whicheach trial comprised a standard and two comparison patterns. Atrandom, one of the two comparisons had the delayed tone. Thus,at the middle tempo, the initial trial with the 22111 pattern and delay of the third tone would take the form shown in Figure 2, althoughthe order of the two comparisons might be reversed.
On each trial, the subject indicated which of the two comparisons contained the delay by pushing one of two keys on a computer terminal. The response interval was 3 sec; if the subject failedto respond in that time, he or she was prompted to answer by amessage on the terminal screen until a response was given. Aftereach response, the terminal screen displayed feedback. In the adaptive procedure, for every two correct answers, the delay was halved;for every incorrect answer, the delay was doubled. For every run,there were five reversals in the amount of detectable delay; eachrun generally took from 12 to 25 trials. The mean and standarddeviation delay detected (in milliseconds) was computed and storedfor the last 10 trials of each run. This procedure results in an estimate of I::J where the probability of a correct response is 70.7%(Levitt, 1971).
The subjects replicated the experiment four times, completingone replication before beginning the next. The order of the 135 runswithin each replication was randomized across patterns, tempos,and position of delay. Data were analyzed for only the last threereplications. Each replication was performed over a period of about
232 MONAHAN AND HIRSH
Table 1Main Effects and Significant Interactions in Experiment 1
Effect df F P
Tempo 2,6 3.33 .107Patterns 8,24 2.92 .020Position of delayed tone 4,12 4.68 .017Replications 2,6 9.74 .013
Tempo x patterns 16,48 2.80 .003Tempo x position of delayed tone 8,24 2.92 .020Patterns x position of delayed tone 32,96 6.29 .0001Tempo x patterns x position of delayed tone 64,192 3.22 .0001
Table 2Mean Difference Limens (in msec) for Main Effects of Experiment 1
Tempo
Slow Medium Fast
14.00 12.06 10.18
Patterns and Position of First Long Interval
DistanceBetween Longs
ZeroOne ShortTwo ShortsM
22111-13.0521211-10.8621121-11.14
11.6&
2
12211-12.6212121-12.3112112-12.08
12.34
3
11221-11.9811212-13.2211211-11.45
12.22
M
12.5512.1311.56
2
13.09"
Position of Delayed Tone
345
Replications (Practice)
2
6
3
12.74 11.88 11.62
Note-Means with different superscripts differ from each other (p < .05) as measuredby Newman-Keuls post hoc tests.
2 weeks with no more than one 2-h session per day. The listenerstypically performed 18 to 20 adaptive runs per 2-h session. Fiveminute breaks were given after every five adaptive runs.
ResultsA repeated measures analysis of variance (ANaVA)
was run on estimatesof the absolute DL for delay (in milliseconds) for the 135 experimental conditions. In a complete factorial design, the analysis had 3 tempos x 9 patterns X 5 positions of the delayed tone X 3 replicationsfor each of the 4 subjects. All main effects except tempowere significant, as shown in Table 1.
Main effects. The means for each main effect, as wellas levels of main effects that differed significantly fromone another, are shown in Table 2. Post hoc analyseswereNewman-Keuls tests with alpha level p < .05 for thewhole experiment.
1. Tempo. Even though the main effect for tempo wasnot significant, the order of tempo means was the sameas that found by Hirsh et al. (1990, Experiment 3) andas that in Experiment 2 (see below). The average absolute DL (~t) was lowest for the fast tempo and highestfor the slow tempo.
2. Patterns. The significant effect of pattern seems tobe related to the high values of ~t in patterns that had2 adjacent long intervals (patterns in row 1 of Figure 1)and in patterns that had a long interval preceding the final tone (11212 and 12112). None of the differencesamong patterns was significantby the Newman-Keulstest.
3. Position ofdelay. Absolute DLs were lowest whenTone 5 was delayed and highest when Tone 2 was delayed(this was the only significant difference that remained soin the Newman-Keuls test); delays of other tones had intermediate effects on DL. This effect may be an artifact,because fewer long intervals abutted Tone 5 than Tone 2in the different patterns.
4. Practice. Absolute DLs improved (became smaller)with each succeeding replication. The effects of practicedid not interact significantly with any other variable.
Interactions. Although the main effects here are ofsome interest, our original focus is better served by ananalysis of the interactions.
First, does discrimination performance depend on theinteraction of pattern (i.e., the position of the long intervals) and the position of tone delay? The answer to thisquestion is yes (p < .0001).
TIMING IN RHYTHMIC PATTERNS 233
Second, given that there is an interaction between pattern and tone delay, does that interaction depend on thetempo of the patterns? The answer to this question is alsoyes (p < .00(1).
Figure 3 shows the interaction of pattern and positionof the delayed tone on the absolute DLs, with temposhown as the parameter. The pattern is indicated by thelength of intervals along the abscissa, so the first patternin Figure 3 is 22111, and so forth. Each subject's performance was averaged over three replications for eachexperimental condition. Each point on the graphs of
Figure 3 represents an average of these averages; the standard error for each point is thus based on an n of 4. Standard errors less than I msec are not shown.
At all tempos, DLs tended to be higher when the delayedtone marked the boundary between 2 long intervals (i.e.,in Patterns 22111, 12211, and 11221) or when Tone 6was delayed and was preceded by a long interval (i.e.,in Patterns 11212 and 12112). At the slow tempo, DLswere generally smallest (performance was best) when thedelayed tone was the boundary between 2 short intervalsand were of intermediate value when the delayed tone was
30,----------------,
25
UillUJE
C
20
15
10
O'---....L__...L-__.L.--'----'_-'---l
+-<J
2 4 5 6 2 3 4 5 6 2 3 4 5 6
30,---------------,
65432
--- .....
INTERVAL<MS) SHORT LONG f----------,
o 0 200 400
.----------. 100 200'V..... . 'V 50 100
432
10
15
25
20
5
35,-----------j
15
30
25
20
10
o-.Io::ICf)
W0:::::If-
Zo0---<
f«Z0---<
L0---<
0:::UCf)0---<
o
45634525 6234oI:..----L__....L---'-_..L-__'----'---'
TON E POSITION
Figure 3. Absolute difference limens (DLs) for delay (.:1t in milliseconds)for nine temporal patterns comprising 6 tones and 5 intervals(top row, 22111, 12211, 11221; middle row, 21211, 12121, 11212; bottom row, 21121, 12112, 11211). In each pattern, Tone 2, 3, 4, 5,or 6 was delayed. The absolute DL, measured at each of three tempos, is shown as the parameter. Each point is the average of 4 subjects'average DL across three replications of conditions. Standard errors are based on an n of 4 subjects; standard errors less than 1 msecare not shown. Absolute DLs are related to tempo: the faster the tempo, the lower the absolute DL.
234 MONAHAN AND HIRSH
the boundary between short and long or long and shortintervals. This result may be interpreted as followingWeber's law, a point to which we shall return.
The results at the medium tempo may not be interpretedas following Weber's law because DLs were almost aslarge for delays between 2 short intervals (1-1 delays) asthey were for delays between 2 long intervals (2-2 delays).
At the fast tempo, absolute DLs were generally higherfor tones delayed between a short and a long interval (i.e.,for 1-2 delays) than for tones between a long and a shortinterval (i.e., for 2-1 delays). This asymmetry, which wasnot present at the slow and medium tempos, contributesgreatly to the three-way interaction oftempo, pattern, anddelay position.
Finally, we note that performance for the control pattern 11211 in the present experiment was similar to thatfound by Hirsh et al. (1990) for the same pattern. Thispattern is heard as short-short-Iong-short-short-long;performance at the slow and medium tempos was poorfor the fourth tone in the series (ending the first long interval and beginning the following short). This may indicate a grouping effect where listeners are quite accuratein hearing timing differences within temporal groups butnot between groups.
We defer further discussion of the results of Experiment I-in particular, measures of relative DL-andWeberian or accent models of discriminability until afterthe results of Experiment 2 have been presented.
EXPERIMENT 2
PatternsThe nine patterns of Experiment 2 were mirror images
of those of Experiment 1 in that long intervals were substituted for short, and vice versa. Thus, the patterns had2 short and 3 long intervals, with the exception of the22122 pattern, which is the mirror image of the 11211pattern of Experiment 1. Figure 4 shows these patternsalong with positions of temporal accent as suggested byPavel and his colleagues. All patterns in the first row ofFigure 4 have regular temporal accenting every 2 grainintervals. Patterns in rows 2 and 3 have more irregularpatterns of accenting; accenting every 3 grain intervalsis most prevalent for the former, and accenting every 2grain intervals is most prevalent for the latter. In addition,the 21212 pattern may be heard as a repeated long-shortpattern with regular accenting (every 3 grain intervals),and the 21221 pattern tends to be heard as a repeatedlong-short-long rhythm, since the interval after the finaltone of the pattern is usually heard as long.
Overview of the DesignThe tempos and positions of tone delay employed in
the present experiment were the same as those employedin Experiment 1. Thus, the designs for the two experiments, except for the use of nine different temporal patterns, were identical-namely, complete factorials with
DISTANCE ONEBETWEEN"SHORTS"
POSITION OF FIRST SHORT INTERVAL
TVO TIRE
< ~ ~ ~ ~
ZERO 1.1.1 2 I 2 I 2 IL"LONGS"~
s s s s
TWO 1.121211121L"LONGS"~
Figure 4. The nine standard 6-lone palterns of Experiment 2. Palterns are mirror-images of those for Experiment 1,so that short intervals are now in the positions previously held by long intervals, and vice versa. Thus, eight of the patterns have 3 long and 2 short intervals and Paltern 22122 is the mirror of Paltern 11211. The levels of natural accentfor each tone are shown according to the rules of Povel and his coUeagues (see text). Palterns are arranged in the figureverticaUy by the distance between the 2 shorts and horizontally by the position of the first short interval.
3 tempos x 9 patterns x 5 positions of delay, totaling135 conditions.
MethodThere were 3 subjects, 1 male and 2 females, with 4, 5, and 10
years of musical experience, respectively. All subjects had normalauditory sensitivity. Two of the subjects, J.B. and H.D., had alsobeen subjects in Experiment 1. J.K., the new listener, like J.B.,was a self-run subject. The subjects were paid for their services.
Both the apparatus and the procedure were the same as in Experiment 1.
ResultsA repeated measures ANOVA was run on estimates of
the absolute DL for delay in milliseconds for the 135experimental conditions. In a complete factorial design, theanalysis had 3 tempos x 9 patterns x 5 positions of thedelayed tone x 3 replications for each of the 3 subjects.Table 3 shows the main effects and significant interactionsof this analysis. Table 4 shows the average for each levelof each main effect for the experiment.
Main effects. In this experiment, tempo was the onlymain effect that was significant. The smallest absolute DLs
TIMING IN RHYTHMIC PATTERNS 235
occurred for the fast tempo, and the largest occurred forthe slow tempo. The same order of tempo means wasfound in Experiment 1 (compare Tables 2 and 4 for average absolute DLs for each tempo). Hirsh et al.'s (1990)Experiments 1 and 3 also had the same order of averageabsolute DLs. This very consistent result implies thatWeber's law does not hold across tempos in the rangewe are examining. IfWeber's law were in effect for thisrange, then the average ~t for the slow-tempo conditionshould be twice that of the medium-tempo condition,which, in turn, should be twice that of the fast-tempo condition. This is clearly not the case.
We note that although the main effect for patterns wasnot significant, patterns in the first row of Figure 4 thathad 2 adjacent shorts or had regular accenting every 2grain intervals averaged a somewhat lower DL than didpatterns in the other rows (see levels of main effects forpatterns in Table 4).
Interactions. The same two interactions that were mosthighly significant in the first experiment-namely, thetwo-way interaction of pattern (the position of the various intervals) with the position of delay, and the three-
Table 3Main Effects and Significant Interactions in Experiment 2
Effect df F P
Tempo 2,4 20.89 .008Patterns 8,16 1.42 .260Position of delayed tone 4,8 1.97 .193Replications 2,4 0.08 .925
Patterns x position of delayed tone 32,64 4.68 .0001Tempo x patterns x position of delayed tone 64,128 2.19 .0001
Table 4Mean Difference Limens (in msec) for Main Effects of Experiment 2
Tempo
Slow
22.19"
Medium Fast
Patterns and Position of First Short Interval
DistanceBetween Shorts
ZeroOne LongTwo LongsM
11222-15.2712122-17.5312212-16.19
16.33
2
21122-14.9921212-16.5221221-16.56
16.02
3
22112-15.1422121-15.1322122-16.83
15.70
M·
15.1316.3916.53
2
17.35
Position of Delayed Tone
345
14.95 15.98 15.28
Replications (Practice)
2
6
16.53
3
15.81 15.98 16.25
Note-Means with different superscripts differ from each other (p < .05) as measuredby Newman-Keuls post hoc tests.
236 MONAHAN AND HIRSH
45
40
35
U 30Q)
25(j)
E 20
C15
f~
10
+- 5
<J 01 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6
45
40
0 35
-.J 300::r:: 25
(f) 20W
15t-----t-~'-
,0::: , ,::r:: 10
l- S v <7<7
Z· 0
0I 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6-I- INTERVAL< MS) SHORT LONG
<t: 45
Z 400 0 200 400
-, .----------. 100 200L 35
V V 50 100- 300:::U 25(f) 20- i;/..f--- /
r--0 15 //
" -. 'f.<7
10 ...
<7 ~.
5 v· ·v0
I 2 3 .. 5 6 2 3 .. 5 6 2 3 .. 5 6
T 0 N E P a s I T I a NFigure 5. Absolute difference limens (DLs) for delay (M in milliseconds) for nine temporal patterns comprising 6 tones and 5 intervals
(top row, 11222,21122,22112; middle row, 12122,21212,22121; bottom row, 12212,21221,22122). In each pattern, Tone 2, 3, 4, 5,or 6 was delayed. The absolute DL, measured at each of three tempos, is shown as the parameter. Each point is the average of 3 subjects'average DL across three replications of conditions. Standard errors are based on an n of 3 subjects; standard errors less than 1 msecare not shown. Absolute DLs are related to tempo: the faster the tempo, the lower the absolute DL.
way interaction oftempo with pattern and position of delay on the absolute DL-were also highly significant inExperiment 2 (compare interactions in Tables 1 and 3).
Figure 5 shows the interaction of pattern and positionof the delayed tone on the absolute DLs, with temposhown as the parameter. The pattern is indicated by thelength of intervals along the abscissa, so the first patternin Figure 5 is 11222, and so forth. Each subject's performance was averaged over three replications for eachexperimental condition. Each point on the graphs ofFigure 5 represents an average of these averages; the standard error for each point is thus based on an n of 3. Standard errors less than 1 msec are not shown.
As in the first experiment, delays between 2 longsyielded a high I1t. At the slow tempo, delays of Tone 6were much easier to detect if they were preceded by ashort interval (e.g., in Patterns 22121 and 21221) thanif they were preceded by a long interval. Also, at the slowtempo, I1twas smallest for delays between 2 shorts; theseoutcomes, as noted above, are in correspondence withWeber's law.
At the medium tempo, delays between 2 shorts werepoorly detected-as was the case in the first study. At thefast tempo, we again found a large disparity in sensitivity to 1-2 and 2-1 delays, with the latter being far betterdetected than the former.
TIMING IN RHYTHMIC PATTERNS 237
A. EXPERIMENT I
3121110
Experiment 2
ns
10II123
Experiment I
(I-I)(1-2)(2-1)(2-2)
Between two shortsBetween long and shortBetween short and longBetween two longs
Delay Context
Table 5Distribution of Delay Contexts
dashed lines. The abscissa in these figures is the averagesize of the 2 intervals within which the delayed toneoccurred. Thus, the average interval size surrounding adelay for both 1-2 and 2-1 delays is 300, 150, or 75 msecfor slow, medium, and fast tempos, respectively. However, a 1-2 delay context means that the order of intervals in the standard pattern was 200-400, 100-200, or50-100 msec, respectively, whereas a 2-1 delay contextmeans the reverse order of intervals. An additional reference (solid line) shows the absolute DL (.Ilt) for the delayof a single tone in otherwise isochronic sequences (Hirshet al., 1990, Experiment 1) where isochronic intervalswere 200, 100, or 50 msec.
To estimate the similarity in outcomes of Experiments Iand 2, we correlated the average absolute DLs for the 12contexts in Figure 6A (4 delay contexts at each of 3tempos for Experiment 1) with the same 12 contexts forExperiment 2 in Figure 6B, and found r(lO) = .804(df = 10, p .s .(01).
At all tempos in both experiments, delays between 2longs (2-2 delays) gave the highest average DLs (seeFigures 6A and 6B). At the slow tempo, tu increased withthe average size of the intervals surrounding the delay.At the medium and fast tempos, this was not the case.The result within slow-tempo patterns is consistent withWeber's law. We also note that our slow-tempo resultsagree with the findings of Bharucha and Pryor (1986),who employed a different procedure for investigating thediscriminability of timing. They increased the length ofsingle intervals by half a beat (and thereby increased totalpattern length) in rhythmic patterns that were made upof relatively random sequences of intervals. They foundthat as t (the size of the interval) increased, .Ilt (the absolute DL) also increased. The size of temporal intervalsin their experiment ranged from 200 to 1,200 msec. However, the absolute DLs for 1-2 and 2-1 delays (averageinterval = 300 msec) at the slow tempo were 7-9 msechigher in our Experiment 2 than in Experiment 1; atpresent, we have no good explanation for this result.
The experiments agree fairly well in levels of the absolute DL at medium and fast tempos, especially in theunpredicted result that 2-1 delays are much better detectedat the fast tempo than are 1-2 delays.
The experiments agree on a further interesting outcome.The delay between 2 shorts at the slow tempo is physically equal, in terms of surrounding intervals, to the delay between 2 longs at the medium tempo. The differencein tu to the same physical delay in the two different tempocontexts in both experiments is slightly above 6 msec. The
1-1." : : 1-1
"il6
<i 12
24
IIIE
025 50 75 100 150 200 300 400 800Average Length of Interval Surrounding Delay (ms)
O~---'---'-----'--.LU.--'-'---'--'---'-----'-----I.----'--'--J
25 50 75 100 150 200 300 400 800Average Length of Interval Surrounding Delay (ms)
6
<i 12
18
18IIIe
Another way of viewing the results. From the abovenoted correspondences, it seemed likely that the resultsof the two experiments would be quite similar if seen asa function of delay context-that is, the intervals surrounding the delayed tone-and tempo.
Figures 6A and 6B show the value of the absolute DLat the three different tempos in the four delay contexts(I-I, 1-2, 2-1, 2-2) in Experiments 1 and 2. The twoexperiments have different numbers of these contexts. Excluding delays of Tone 6, there are 36 delay contexts ineach experiment divided among the four types as indicated in Table 5. Standard errors for each point inFigure 6 are based on the number of contexts for eachdelay type; standard errors less than I msec are notshown. Contexts within the same tempo are connected by
24
Figure 6. The value of the absolute difference limen (DL; ~t) atthree different tempos and in four delay contexts (I-I, 1-2,2-1,and 2-2; seetext) for Experiments 1 (Panel A) and 2 (Panel B). Standard errors for each point in the figures are hased on the numberof occurrences of the four contexts in each experiment (see text);standard errors less than 1 msec are not shown. Contexts withinthe same tempo are connected by dashed lines. The abscissa is theaverage size of tbe 2 intervals within which tbe delayed tone occurred.An additional reference (solid line) is provided, showing tbe absoluteDL for the delay of a single tone in otherwise isochronic sequences(Hirsh et al., 1990, Experiment 1), where isochronic intervals were200, 100, or 50 msec.
238 MONAHAN AND HIRSH
DISCUSSION
025 50 75 100 i 50 200 300 400 800Average Length of Interval Surrounding Delay I msI
standard context for these delays contains 2 consecutive200-sec intervals. However, at the medium tempo, thedelay sounds as if it occurs between temporal groupings(between 2 longs), whereas at the slow tempo, the delaysounds as if it occurs within a temporal grouping (between2 shorts).
ferent experiments, respectively. We are investigatinghere the boundary where Weber's ratio for discrimination of time begins to be constant for the comparison ofsingle intervals, apparently somewhere between 100 and200 msec (see Killeen & Weiss, 1987, for a recentreview). In fact, i1tft short or long increases with increasingtempo or decreasing size of either the shorter or the longerinterval; this is in general agreement with the literaturefor judgments of the lengths of single intervals, whichshows that the Weber ratio increases monotonically andsteeply with decreasing interval lengths under 100 msec.
Using a constant value for t at each tempo carries theassumption that a clock-generated beat or pulse is usedfor making discriminations, whether a tone is presentmarking the onset of that beat interval or not. However,the implementation of this model, especially at the slowtempo, implies that the delays of isolated tones, tones followed, and in some cases preceded, by long intervals("accented" tones in Povel & Essens's, 1985, terms),have higher relative thresholds for temporal discrimination (see slow-tempo values in Figures 7A and 7B). Onthe other hand, on the basis of research with comparisonof single temporal intervals, we might expect a constantWeber ratio within the slow tempo. Both Getty (1975)and Kristofferson (1980) have reported a constant ratiobetween .05 and .06 for single intervals of 200 and400 msec, which are, respectively, the short and long ofour slow-tempo patterns.
A Weber Average-Interval ModelAnother version of the Weber hypothesis assumes that
the listener employs both the preceding and the following intervals in making a discrimination. Thus, the t inthe denominator of the Weber ratio is the average of the2 intervals separated by the delayed tone.
Figures 8A and 8B show the Weber ratio calculated onan averaged t for each of the four delay contexts at eachof the three tempos for Experiments 1 and 2. The factthat data for the four contexts at the slow tempo fall ina horizontal straight line, especially for Experiment 1 (seeFigure 8A), suggests that this type of model will providea good fit. In both Figure 8A and Figure 8B, the distribution of the 12 points for I1tft average interval forms acurve that is quite reminiscent of that for I1tft for singleintervals, with two possible exceptions, for which we canpresent only ad hoc explanations.
First, in both experiments, according to the I1tft average measure, delays were much better discriminated between 2 shorts at the slow tempo than between 2 longsat the medium tempo, despite the fact that both delays hadthe same sizes of surrounding interval. We suggestedabove that this result had to do with whether the intervals in question were within-groups temporal intervals orbetween-groups intervals. Povel (1981) has observed aperceptual difference between the two types of interval.In his study, listeners reproduced sequences with alternating short and long intervals, which he called t, and t 2 ,
respectively. He states:
- 12.5 gc.
- 10.0 ;;-
-7.5 g
- 15.0 ~
-5.0
-2.5
EXPERIMENT 2
45 r-'1r--r--r--r-..........,...,---r---r--r--r--r-,....,22.5
20.0
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40 I- A. EXPERIMENT I - 20.0
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25 I- ", " 12.5 0't ,: A'I~2 MEDIUM :J
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025 50 75 100 150 200 300 400 800
Average Length of Interval Surrounding Delay lmsl
5-
oo
_ 151-<,
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25 I-L,g 20 I-
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A Weber Fixed-Beat ModelOne reasonable choice for a reference standard in the
Weber ratio is t = length of the short or long interval (themost prevalent interval). If the Weber ratio were constantacross tempos, then the values at each of the three temposshould have the same vertical position in Figures 7A and7B, which show relative sensitivity (I1tft short and I1tftlong) for delay in four different contexts in the two dif-
Figure 7. Value of the relative difference limen (the Weber ratio =MIt "short" or MIt "long," Ieft- and right-band vertical scales, respectively, in each panel) at three tempos and in four delay contexts (1-1,1-2,2-1, and 2-2; see text) for Experiments 1 (Panel A) and 2(Panel B). Standard errors for each point in the figures are basedon the number of occurrences of the four contexts in each experiment(see text); standard errors less than 1 I1tlt unit are not shown. Contexts within the same tempo are connected by dashed lines. Theabscissa is the average sizeof the 2 intervals within which the delayedtone occurred.
TIMING IN RHYTHMIC PATTERNS 239
Initial DifferentComparison
75-25 13.3-4075-75 13.3-13.3
125-25 8-40125-75 8-13.3
Period (msec) Rate (Hz)
Standard
50-50 20-2050-100 20-10
100-50 10-20100-100 10-10
Period (msec) Rate (Hz)
1-11-22-12-2
DelayType
Table 6Intervals Surrounding the Position of the Delayed Tone
posed to be stronger than backward masking (Elliott,1962). However, this result is correlated with a perceptual phenomenon of an apparent pitch and/or timbralchange that seems to accompany 2-1 delays but not 1-2delays. At the fast tempo, the standards for trials with2-2 delays have a repetition rate of 10 notes/sec, or10 Hz, surrounding the position of the tone to be delayed;standards for trials with 1-1 delays have a repetition rateof 20 Hz surrounding the position of the tone to bedelayed. In Table 6, we show the standard and initialchanged comparison intervals for the four types of delaycontext at the fast tempo; the intervals are shown bothin terms of their periods (lOIs in milliseconds) and interms of their repetition rates (Hz). The initial delay inthe changed comparison at the beginning of any adaptiverun is 25 msec. Listeners easily notice the large difference in the repetition rate for 2-1 contexts and, to a lesserdegree, for 1-1 contexts. The different comparison for1-2 contexts initially exhibits no difference at all in repetition rate. It is possible that we have tapped into whatBharucha and Pryor (1986) have termed an "asymmetryin the detection of alterations to a sequence, as a functionof whether coherence is violated or established" (p. 137).Thus, it may be argued that the initial different comparisons for the 2-1 and 1-1 contexts at the fast tempo aremore "incoherent" than their respective standards, andhence are easy to detect; on the other hand, the initialdelayed comparison for a 1-2 context is just as "coherent" as the standard, because the intervals surroundingthe delayed tone are equal in length and repetition rate.
An Accent Model Compared witha Weber's Law Model
The smallest lOis for which Povel and Okkerman(1981) report natural temporal accent are about 100 msec;thus, it is probably inappropriate to apply a temporal accent model to our fast-tempo patterns. It is also clear fromthe preceding section that a Weber's law model is notlikely to fit data from within patterns at either the fastor the medium tempo. Therefore, we will compare thetwo models only at the slow tempo as predictors of the36 absolute DLs we obtained in each experiment (9 patterns X Delay Positions 2,3,4, and 5); we will then compare the two models for the 72 DLs from the combinedexperiments.
Table 7 shows the predictions of the two differentmodels for the discriminability of delay for each of fourtone positions (2,3,4, and 5) for each pattern in Experi-
-
-
-
-
-
EXPERIMENT 2
EXPERIMENT I
I
B.
A.
025 50 75 100 150 200 300 400 800Average Length of Interval Surrounding Delay I ms)
QI 18 I
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If a t,-1, pattern is presented to a subject, beeps and not intervals will be perceptually grouped according to their proximity. Thus t, becomes part of the group (figure) and 1, partof the background.... Between-groups intervals are special.Subjects indicate that such intervals have hardly any reality for them. Indeed, in imitating these sequences one sensesthat much attention is directed toward imitating the withingroup interval, whereas the between-groups interval hardlyrequires any attention. (p. 10)
Thus, it is possible that, because of the different attentionaldemands of the two contexts, our listeners found it easierto hear the same delays in a within-groups context (between 2 shorts at the slow tempo) than in a between-groupscontext (between 2 longs at the medium tempo).
Second, at the fast tempo, in both experiments, 2-1 delays were much easier to detect than were 1-2 delays.One possible explanation is that the delayed tone is closerto the preceding tone in the 1-2 context and is thereforemore subject to forward masking, which is generally sup-
50 75 100 150 200 300 400 800Average Length of Interval Surrounding Delay (ms l
Figure 8. Value of the relative difference limen (the Weber ratio =J1tlt"average") at three tempos and in four delay contexts (1-1, 1-2,2-1, and 2-2; see text) for Experiments 1 (Panel A) and 2 (Panel B).Standard errors for each point in the figures are based on the number of occurrences of the four contexts in each experiment (see text);standard errors less than 1 tJ.tlt average unit are not shown. Contexts within the same tempo are connected by dashed lines. The abscissa is the average size of the 2 intervals within which the delayedtone occurred.
240 MONAHAN AND HIRSH
Note-Patterns are shown with levels of natural accent. ::s = strongtemporal accent. < = moderate temporal accent. *Weber modelweights are the mean of 2 intervals around the delayed tone dividedby 100.
SUMMARY
We have found that trained listeners can discriminatea delay in the timing of a tone in series of tones about
(Experiment 4) played repeating sequences consisting ofalternating long and short intervals. As noted above, thetones that are followed by the long interval in such a seriessound accented. In their Experiment 4, Povel and Okkerman had their subjects raise the level of the unaccentedtones in the series (tones followed by short intervals) untilthey sounded as loud and important as the temporally accented tones. In general, the subjects had to raise the levelby about 4 dB. Thus, we assume that the relative levelsof accent for strongly accented and unaccented tones inthe model differ by 4 dB. We have arbitrarily chosen 0and 4 dB as low and high values of accent, and employed2 dB as the middle value, which is evenly interpolatedbetween 0 and 4.
The Weber averaged-interval model, on the other hand,predicts that At is proportional to the average of the lengthsof the 2 intervals surrounding the delayed tone. The slowtempo average interval for 1-1 delays is 200 msec, for1-2 or 2-1 delays is 300 msec, and for 2-2 delays is400 msec. These average values, divided by 100, givethe weights for the Weber model shown in Table 7.
Three types of correlation are shown in the top portionof Table 8: (1) Model weights are correlated with eachother for each experiment and for both the accent and theWeber models; (2) model weights for both models arecorrelated with experimental results (At) for the 36 conditions of Experiment 1 and the 36 conditions of Experiment 2; and (3) At for the 36 conditions of Experiment 1is correlated with At for the 36 conditions of Experiment 2.
As can be seen from the table, the Weber averagedinterval model predicts At better for both experiments thandoes the accent model, and, furthermore, it accounts formore than half the variance (R2 = .594) of the 36 pointsfrom Experiment 1. The Weber model, without involving the estimation of any parameters at all, does a goodjob of fitting the data from Experiment 1 and a mediocrejob of fitting the data of Experiment 2. The accent modelfits Experiment 1 data slightly better than it does thoseof Experiment 2, but the fit to both data sets is poorat best.
We note that the predictions of the two models are significantly correlated (r = .642) across the the 72 conditions of the two experiments. The partial correlation ofthe Weber model predictions with At from the 72 conditions from both experiments-holding the predictions ofthe accent model constant-is .529 (df = 70, P < .(01).On the other hand, the partial correlation of the predictions of the accent model with At from the same 72 conditions from the two experiments-holding predictions ofthe Weber model constant-is .07 (df = 70, p = n.s.).Thus, we may conclude that the accent model predictsvery little of the variance in discrimination behavior independently from what is predicted by the Weber model.
4
4
3
4
3
2
3
2
3
234 3
3 3 3 3
3 3 3 2
3 4 3 2
2 3 4 5
233 3
4 3 2 2
3 3 2 3
2
3 2 3 3
3
3 3 3 3
3 3 3 4
2 3 3 2
4 3 3 3
3 4 3 3
3 4 3 3
4
4 3 3 4
4
4
4
o 4 4 0
2 040
o 4 2 0
o 4 0 4
4 2 0 4
2 345
4 4 2 0
204
4 0 4 4
4 040
4 0 4 0
420
o 4 4 0
o 4 0 4
o 4 2 0
4 4 0 4
Experiment I
4 2 0 0
Experiment 2044
:S S
6. I 1 2 . I 2 . L<::s ::s::s
7.2·112·IL::s <::s ::s
8. I 2 . I I 2 . L::s < ::s ::s
9.112·IIL.< ::s < -s
Pattern
1.2·2·IIIL::s ::s < ::s
2. I 2 . 2 . I I L::s ::s < ::s
3.112·2·IL< ::s::s ::s
4. 2 . I 2 . I I L::s ::s < ::s
5.12·12·IL
1.112·2·2·L< ::s ::s ::s ::s
2 . -2 . I I 2 . 2 . L::s < ::s ::s ::s
3.2·2·112·L::s ::s < ::s ::s
4. I 2 . I 2 . 2 . L
5.2·12·12·L
7.12·212·L
6. 2 . 2 . I 2 . I L
8.2·12·2·IL
9. 2 . 2 . I 2 . 2 . L 4 0 4 4
Table 7Model Weights Predicting .1t for Four Positions of Tone Delay
(Tones 2, 3, 4, and S) in 18 Patterns from the Two Experiments
Accent Model Weber ModelWeight Weight*Position Position
ments 1 and 2. The accent model that we test here predictsthat temporal sensitivity is inversely related to the levelof perceived temporal accent as described by Povel andOkkerman (1981) and Povel and Essens (1985). Themodel provides an ordinal scale of three levels of temporal accent: (1) Delays of tones that initiate long intervals, marked v s " in Table 7 (the markings are identical to those in Figures 1 and 4), will be poorly detected(At will be large); (2) delays of tones that begin a seriesof 3 or more intervals, marked " < " in Table 7, willbe moderately well detected; and (3) delays of tones thatare unmarked by temporal accent will be best detected(At will be small). The initial weights of the accent modelwere not chosen entirely arbitrarily. Povel and Okkerman
TIMING IN RHYTHMIC PATTERNS 241
Table 8Weights of Two Models Correlated with I1t for Conditions of Experiments 1 and 2
Weber Model Accent ModelExp. I
Difference Limen
Exp. I Exp.2 Exp. I Exp.2 (t.t)
Correlations for Two Experiments Separately
-1.00*.676* - .676*
- .580* .580* - .898*
Weber Model: Exp. IAccent Model: Exp. IAccent Model: Exp. 2
Exp. I (I1t)Exp. 2 (I1t)
.771*-.441*
-.771*.441*
.514*-.396t
-.35lt.383t -.360t
Correlations for Combined Experiments
Accent Model .642*Combined Exps. (I1t) .603* .433*
Note-Correlations for two experiments separately: n = 36 conditions, df = 34 for each correlation.Correlations for combined experiments: n = 72 conditions, df = 70 for each correlation. *p < .01,two-tailed test. tp < .05, two-tailed test.
as well as they can when such tones define single temporal intervals, according to an older body of literature.When the sequences include combinations of long andshort intervals, thereby generating rhythmic structuresmore complex than mere equal timing, there are someinternal dependencies. For example, at slow tempos(short, long = 200, 400 msec), a tone's temporal position is more easily discriminated when it separates 2 shortintervals than when it separates any other combination ofshort and/or long intervals. At all tempos, discriminationfor tones between 2 long intervals is difficult. At the fasttempo (short, long = 50, 100 msec), there is an interesting asymmetry for the results on tones separating a shortand a long interval. Delays of tones following long andpreceding short intervals are better discriminated thanthose following short and preceding long intervals.
In addition to the model suggested above, we have attempted other models based on accent structure. None ofthem predicts the slow or musical tempo performance aswell as does the average-interval Weber's law model. Themajor point is that temporal discrimination performancein complex, quasimusical contexts may well be modeledby an older psychophysical principle, at least for thesenoncycled sequences. It may be that appealing to cognitive representations of meter, accent, and the like willbe unnecessary.
REFERENCES
ABEL,S. M. (1972). Duration discrimination of noise and tone bursts.Journal of the Acoustical Society of America, 51, 1219-1223.
BHARUCHA, J. J., 8t PRYOR, J. H. (1986). Disrupting the isochrony underlying rhythm: An asymmetry in discrimination. Perception & Psychophysics, 40, 137-141.
BOLTZ, M., 8t JONES, M. R. (1986). Does rule recursion make melodies easier to produce? If not, what does? Cognitive Psychology, 18,389-431.
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(Manuscript received January 13, 1989;revision accepted for publication September ll, 1989.)